2.1. Franck-Condon factors

The Franck-Condon factor is the square of the integral over the product of the vibrational eigenfunction of the two states involved (the so-called overlap integral). It controls the intensity distribution from band to the band across a system. Franck Condon factors and r-centroids play an important role in finding out the variation of electronic transition moment with internuclear distance [⁵,⁶]. The principle based on the electronic jumps is faster compared to the nuclear motion. This principle directs us to see that the bands are intense if the jump from a higher vibrational level to a lower level takes place from either the r _min or r _max position on the potential energy curves’ straight to lower level, because then there is a minimum or zero change in position, i.e., ∆r(∆r = r _e’ − r _e’’ ) and in momentum is possible. In emission, for one v’ value, there are two values of v”. The locus of such points on the Deslandres table called the Condon parabola [⁵]. If the Condon parabola is wide open, then bands of many progressions and sequences appear in a band system. On the other hand, if the Condon parabola is narrow and tends to degenerate in a diagonal line in the Deslandres table along the main sequence, then only the bands of the 0-0 sequence or the 0-1 or 1-0 sequences will hardly appear. The intensity depends on many other factors, such as the probability of transition, the population of the upper state from where the transition is taking place, the line strength or band strength (depending on the rotational line or vibrational band), and the electronic transition moment.

In emission, the expression is given as:

where c is the velocity of light, N _v’ is the number of molecules in the v’ ^th level, υ is the frequency of radiation, R¯e is the average value of Re (i.e., electronic transition moment), and Ψ _v’ and Ψ _v’’ are the wave functions of the v’ and v” levels, respectively.

In absorption, the expression is given by:

where h is the Planck’s constant, I ₀ is the initial intensity, ∆x is the thickness of the medium, and N _v” is the number of molecules in the v” level. Re¯ and R _e are related to each The Franck Condon principle, which governs the intensities of the bands based on the assumption that the variation other by the expression:

The Franck Condon principle, which governs the intensities of the bands based on the assumption that the variation of Rev',v″ with r, i.e., internuclear separation is slow. The Franck Condon principle also states that the electron jumps in a molecule take place so fast in comparison to the vibrational motion that immediately afterward the nuclei still have very nearly the same relative position and velocity as before the jump. In short, the bands obeying this principle are strong. If the minima of the potential energy curve of the upper electronic state and the lower electronic state lie above one another or re'∼re″ the 0-0 sequences may likely appear with strong intensities and other sequences like ∆v = ±1, etc. may not be seen at all. If re'>re″ or minima of the upper PE curve is displaced slightly with respect to minima of lower PE curve then bands of the same intermediate sequence may be stronger. If re'≫re″, then the maxima of intensity may lie in the continuum. In the case of emission, there are usually two maxima in v ⁰⁰ progressions because molecule in the upper electronic state can be in any excited vibrational level and jumps can occur from either turning points on PE curve. Details are given in Herzberg [⁵], and Straughan and Walker [⁶].

The quantity in the square bracket appearing in Eq. (1) and (2) is known as Franck Condon factor usually denoted by q_v’v’’ and expressed as

There are different methods of calculating Franck-Condon factors. Expressions are not simple, as the expression for wave-functions Ψ _v’ and Ψ _v’’ are very complex and sometimes may involve special functions. Chakraborty and Pan [⁷] have surveyed these methods Sharp [⁸]. Telle et al. [⁹] have also given the methods to calculate Franck-Condon factors. The comparative study of the FCFs and r-centroids carried out by Nadhem et al., [¹⁰,¹¹] and Ramon S. da Silva et al., [¹²]. In the present study, we used a program developed by Jarmain and McCallum [¹³,¹⁴] to calculate these factors. In this program, the Klein-Dunham series is used to represent the spectroscopic input data, namely, the vibrational and rotational constants. The RKR potential curves are then computed. The Schrodinger wave equation is solved¨ numerically, and the resulting vibronic eigenfunctions are used to calculate the Franck Condon factors and r-centroids [¹⁰,¹¹,¹⁵].

2.2. r-centroids

Various methods of calculation of r-centroids are described by Jarmain and Nicholls [¹⁴]. r-centroids is defined as:

Where Ψ1v'(r) and Ψ2v″(r) are the vibrational wave functions of the level v’ and v’’. If these are normalized, then the denominator of Eq. (5) is equal to unity. The methods for the calculation of r-centroids are described by Jarmain and Nicholls [¹⁴], which are (i) direct method, (ii) quadratic equation method, and (iii) difference method. In the present study the direct numerical method is used for the computation of r-centroids.

2.3. Molecular constants

The vibrational and rotational constants of the A2Π _i , B²∑⁺ and D²∑⁺ states are summarized in Table I and are derived from the experimental studies [¹⁶-²⁰]. The Franck-Condon factors and r-centroids of the D-A and D-B band systems are computed and presented in Tables II and III, respectively.